Braid Group

(i) Consider the braid group $\mathcal{B}_{5}$ with the generators $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$, $\sigma_{4}$. Simplify

(ii) Consider the braid group $\mathcal{B}_{3}$ and with the generators $\sigma_{1}$ and $\sigma_{2}$. Let $a = \sigma_{1}\sigma_{2}\sigma_{1}$, $b = \sigma_{1}\sigma_{2}$. Show that $a^{2} = b^{3}$.

Consider the braid group $\mathcal{B}_{3}$. A faithful representation for the generators $\sigma_{1}$ and $\sigma_{2}$ is

Both are elements of the group $SL(2,\mathbb {Z})$ and the Lie group $SL(2,\mathbb {R})$.

Consider the unitary $2 \times 2$ matrices

where $\theta \in \mathbb {R}$. Find the conditions on $\theta_{1}, \theta_{2}, \theta_{3} \in \mathbb {R}$ such that (braid-like relation)

Can one find $2 \times 2$ matrices $A$ and $B$ with $\left[A,B\right] \neq 0_{2}$ and satisfying the braid-like relation $ABBA = BAAB$?

Do the $2 \times 2$ unitary matrices

satisfy the braid-like relation $ABA = BAB$?

The $3 \times 3$ matrices $g_{1}$ and $g_{2}$ play a role for the matrix representation of the braid group $\mathcal{B}_{4}$

as generators. Let $g_{2} = g_{1}g^{-1}$. Find the eigenvalues and eigenvectors of $g_{1}$, $g_{2}$.

Find the conditions on $a,b,c,d,e,f \in \mathbb {R}$ such that

Find nontrivial solutions to these conditions.

Let $A$, $B$ be $n \times n$ matrices with $ABA = BAB$.

Let $T$ be an $n \times n$ matrix and $R$ be an $n^{2} \times n^{2}$ matrix. Consider the equation

Find all $4 \times 4$ matrices $A$, $B$ with $\left[A,B\right] \neq 0_{4}$ satisfying the conditions

The first condition is the braid relation and the second condition $ABBA = I_{4}$ runs under Dirac game.

The free group, $\Gamma_{2}$, with two generators $g_{1}$ and $g_{2}$ admits the matrix representation

Let $A$, $B$ be $2 \times 2$ matrices and

Find the conditions on $A$ and $B$ such that $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$.

Let $I_{2}$ be the $2 \times 2$ unit matrix. Let $R$ be a $4 \times 4$ matrix over $\mathbb {C}$. Then $R$ satisfies the Yang-Baxter equation if

Use computer algebra to show that

satisfy the Yang-Baxter equation.

Show that the $2 \times 2$ matrix

satisfies the braid-like relation

Show that the $3 \times 3$ matrix

satisfies the braid-like relation

Consider the $2 \times 2$ matrices $I_{2}$ and $\sigma_{1}$ and the $4 \times 4$ matrices

Calculate the matrix $\widetilde{R} = FR$. Find the determinant of $\widetilde{R}$. Does the $4 \times 4$ matrix $\widetilde{R}$ satisfy the braid-like relation

Consider the four Bell states

Apply the matrix $\widetilde{R}$ to the Bell states. Discuss.

Let

and $B(k)$ the $2^{k} \times 2^{k}$ matrix recursively defined by

Let

Then $BCB = CBC$. Is

Let $n \geq 2$. Let $u,v \in \mathbb {C}$ ($u$, $v$ are called the spectral parameters). Consider the $n^{2} \times n^{2}$ matrices $R(u)$ over $\mathbb {C}$ with the entries $(r_{k\ell }^{ij})(u)$ with $i,j,k,\ell = 1,\ldots ,n$ and the entries are smooth functions of $u$. Let $\left\{ \overrightarrow {e}_{i} : i = 1,\ldots ,n\right\}$ be the standard basis in the vector space $\mathbb {C}^{n}$. Let $\otimes$ be the Kronecker product. Then

is the standard basis in the vector space $\mathbb {C}^{n} \otimes \mathbb {C}^{n}$. One defines

Thus

For $n=2$ one finds that the matrix $R(u)$ is given by

One defines the three $n^{3} \times n^{3}$ matrices

and

Let $q=4$ and $w := \exp (2\pi i/q)$, i.e. $w^{4}=1$ and $w=i$, $w^{2}=-1$, $w^{3}=-i$. Let $D_{1}$ and $D_{2}$ be the diagonal matrices

and $P$ the permutation matrix

Consider the $64 \times 64$ matrices

Show that

where $0_{64}$ is the $64 \times 64$ zero matrix.

Consider the Pauli spin matrices $\sigma_{0} = I_{2}$, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$. Find the $4 \times 4$ matrix

with

where $\eta$ is a fixed parameter.

Let $\mathcal{B}_{n}$ denote the braid group on $n$ strands. $\mathcal{B}_{n}$ is generated by the elementary braids $\left\{ b_{1}, b_{2}, \ldots , b_{n-1}\right\}$ with the braid relations

Let $\left\{ \overrightarrow {e}_{1}, \overrightarrow {e}_{2}, \ldots , \overrightarrow {e}_{n}\right\}$ denote the standard basis in $\mathbb {R}^{n}$. Then $\overrightarrow {u} \in \mathbb {R}^{n}$ can be written as

Consider the linear operators $B_{j}$ ($\alpha ,\beta ,\gamma ,\delta \in \mathbb {R}$ and $\alpha ,\gamma \neq 0$) defined by

and the corresponding inverse operation

Use computer algebra to show that $B_{1}, B_{2}, \ldots , B_{n-1}$ satisfy the braid condition

if $\gamma \beta +\delta = \alpha \delta +\beta$.

Let $n \geq 2$ and $j,k = 1,\ldots ,2n-1$. Let $Q$ be the $4 \times 4$ matrix

Then $Q^{2} = (s^{4}+s^{-4})Q$. Consider the $2^{2n}$ matrices

where the $4 \times 4$ matrix $Q$ is at the $j$-th position. These matrices satisfy the relations (Temperley-Lieb algebra)

These matrices play a role for the $q$-state Potts model. Consider the case $n=2$ and the $16 \times 16$ matrices

Write computer algebra programs in SymbolicC++ and Maxima which check that these matrices satisfy

Consider the $4 \times 4$ matrices

Is $AB^{-1}A = B^{-1}AB^{-1}$?

Let

be an element of the Lie group $SL(2,\mathbb {R})$, i.e. $\operatorname {det}\left(T\right) = 1$ and $t_{jk} \in \mathbb {R}$. Let

Find all $T \in SL(2,\mathbb {R})$ such that $\left[R, T \otimes T\right] = 0_{4}$.

Let $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ be the Pauli spin matrices. Consider the $4 \times 4$ matrix

where

Does $R$ satisfy the braid-like relation

Consider the braid group $\mathcal{B}_{3}$ with the generators $\left\{ \sigma_{1},\sigma_{2}\right\}$ and the relation

Let $t \neq 0$. Show that a matrix representation is given by

with the inverse matrices

Let $\sigma = \sigma_{1}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}^{-1}\sigma_{1}^{-1}$. Find $f(t) = \operatorname {det}\left(\sigma - I_{2}\right)$. Find minima and maxima of $f$.